About Linear Algebra
What linear algebra is about, what students learn, why it matters, and how I taught it.
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The main idea
Linear algebra studies vectors, matrices, systems of linear equations, and linear transformations. These ideas give us a compact way to represent many quantities at once and to describe how they are combined or changed.
A matrix is therefore more than a table of numbers. It can describe a system of equations, rotate an object, mix information, or transform a large set of data. Learning linear algebra means learning to see these different uses as parts of the same structure.
What students learn
Students begin with vectors and matrices, then study row reduction, systems of linear equations, determinants, linear combinations, bases, inner products, eigenvalues, and eigenvectors. The course also connects these ideas to finite fields, transformations, graphics, encryption, and data.
The weekly materials were organised around questions such as “What is a vector?”, “Why do we need matrices?”, and “How do we solve a system of linear equations?” This keeps definitions connected to a reason for learning them.
Why it matters
Linear algebra is one of the main mathematical languages used by computers. Images, three-dimensional objects, encrypted messages, and datasets can all be represented by vectors and matrices. Matrix operations then make it possible to move a 3D object, encrypt text, reduce data, or train a model.
It is also useful because the same small set of ideas works across many subjects. A student who understands vectors, transformations, and eigenvectors can recognise them again in graphics, machine learning, differential equations, and quantum computing.
How the course worked
The course used questions, reading, writing, discussion, and programming together. An early project used the Advanced Encryption Standard to give matrices and finite-field operations a clear purpose. A larger project asked students to make an interactive 3D program in Go using the Ebiten game package. They built the transformation themselves rather than using a ready-made 3D graphics package.
The 3D project grew in stages: rotate and move an object, add shading, create mathematical texture patterns, and improve the user experience. Groups then presented executable programs in a poster-style fair, tried one another’s work, gave feedback, and explained the mathematical choices behind their designs. The finished program was important, but so were the discussion, written explanation, and ability to connect each visual effect to a matrix operation.